1. ASEN 5070: Statistical Orbit Determination I Fall 2015
Professor Brandon A. Jones
Lecture 39: Combining State Estimates
and Measurement Modeling

2. Announcements/Reminders
No lecture quiz next week.
Exam 3 posted today by 5pm
In-class Students: Due December 11 by 5pm
CAETE Students: Due 11:59pm (Mountain) on 12/13
Final Project Due December 14 by noon
Eduardo’s office hours next week:
He will be out of town Thursday and Friday, but is available via .
Office hours Thursday (2:30-4:30pm) have been moved to Tuesday (2:30-4:30pm in ECAE 1B44)

3. Exam 3 Comments
Your solutions must be uploaded to D2L as a searchable PDF
Same rules as homework apply in regards to format, code appendices, etc.
Open-book, open notes
You may use a computer, MATLAB, etc.
Honor code rules apply
Do not give or ask for help from your peers
The TA has been instructed to redirect all questions to the instructor
I can answer questions to clarify what is being asked, but cannot provide guidance on solutions

4. Project Q&A

5. Combining State Estimates

6. Consider This Scenario
A ground station in Maui observed our satellite several times over the past week
Generated a filtered solution using their observations
A ground station in Florida also observed the satellite several times over the past week
Generated a filter solution using their observations
What is the best approach to fusing this information?

7. Solution Setup
Treat one solution as the a priori and the other as the observation
Does it matter which one is which?
For this case, H=I

8. Combining Solutions
Does not require the additional processing of observations

9. General Combination of Solutions

10. Modeling Measurements Tapley, Schutz, and Born, Chapter 3
Montenbruck and Gill, Satellite Orbits, Chapter 6

11. Range Types One-way Range Example: GNSS
Signal travels to/from reference from/to satellite

12. Range Types Two-way Range Examples: SLR, DSN
Satellite is a relay for signal

13. Range Types Multi-way Range Examples: DSN, TDRSS
Multiple satellite and/or ground stations used

14. Ideal Range and Range-Rate
We have been using range and range-rate:
In the real world, what is wrong with these equations?

15. Light Time Correction At best, a signal travels at the speed of light
We must approximate the signal propagation time δt
Approximately 0.06 seconds for GPS signal to reach Earth
A LEO spacecraft will have moved approximately 500 meters in that time

16. Light Time Computation
Assume we have estimates of our satellite trajectory and the reference station/satellite
We need to solve for δt
No analytic solution so we solve for the correction using iteration

17. Light Time Correction Algorithm
Start with δt=0
Compute the distance with the satellite state at time t and the reference state at t-δt
Given that distance, compute the light propagation time Δδt
Set δt=δt+Δδt
Continue until Δδt is sufficiently small

18. Any other issues?
We have taken care of light-time correction assuming the speed of light in a vacuum.
Any other things we should account for?
Signal does not always propagate through a vacuum
Ionosphere
Troposphere
Charged particle interactions
Solar corona
etc.
Coordinate and time systems
This requires a very careful treatment in the filter

19. Coordinate Systems
Not including accurate coordinate system information creates systematic errors.
Violates our assumption of random errors!
Creates a time-varying bias in the measurement
Table courtesy of Bradley, et al., 2011

20. Doppler as Range-Rate
Is it possible to measure range-rate instantaneously?
No! (at least not that I am aware of)
We have to observe this indirectly
Instead, we look at the change in a signal over time to approximate the range-rate

21. Pulsed Transmission
Satellite sends pulse at fixed interval

22. Repeat Pulse Transmission

23. Doppler Shift
Image Courtesy of WikiCommons
The velocity of the spacecraft affects the frequency of any radar signal
Requires us to observe the change in frequency over some period of time
Known as integrated Doppler shift

24. Doppler Shift
Image Courtesy of WikiCommons
The velocity of the spacecraft affects the frequency of any radar signal
Requires us to observe the change in frequency over some period of time
Known as integrated Doppler shift

25. Two-Way Doppler Measurements
Range-rate model assumes:
Linear change in range over integration time
Constant transmission frequency over integration time

26. Doppler Measurements (one model)
Do I need to perform any light time correction?
Is there anything different about this case when compared to range?

27. FCQs